A result of Vietoris states that if the real numbers \(a_1,\ldots ,a_n\) satisfy

$$\begin{aligned} \text{(*) } \qquad a_1\ge \frac{a_2}{2} \ge \cdots \ge \frac{a_n}{n}>0 \quad \text{ and } \quad a_{2k-1}\ge a_{2k} \quad (1\le k\le n/2), \end{aligned}$$

then, for \(x_1,\ldots ,x_m>0\) with \(x_1+\cdots +x_m <\pi \),

$$\begin{aligned} \begin{aligned} \text{(**) } \qquad \sum _{k=1}^n a_k \frac{\sin (k x_1) \cdots \sin (k x_m)}{k^m}>0. \end{aligned} \end{aligned}$$

We prove that \((**)\) (with “\(\ge \)” instead of “>”) holds under weaker conditions. It suffices to assume, instead of \((*)\), that

$$\begin{aligned} \sum _{k=1}^N a_k \frac{\sin (kt)}{k}>0 \quad (N=1,\ldots ,n; \, 0<t<\pi ), \end{aligned}$$

and, moreover, \((**)\) is valid for a larger region, namely, \(x_1,\ldots ,x_m\in (0,\pi )\).

中文翻译：

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具有更多变量的正弦和的不等式

Vietoris 的结果表明，如果实数\(a_1,\ldots ,a_n\)满足

$$\begin{aligned} \text{(*) } \qquad a_1\ge \frac{a_2}{2} \ge \cdots \ge \frac{a_n}{n}>0 \quad \text{ and } \quad a_{2k-1}\ge a_{2k} \quad (1\le k\le n/2), \end{aligned}$$

然后，对于\(x_1,\ldots ,x_m>0\)和\(x_1+\cdots +x_m <\pi \)，

$$\begin{aligned} \begin{aligned} \text{(**) } \qquad \sum _{k=1}^n a_k \frac{\sin (k x_1) \cdots \sin (k x_m) {k^m}>0。\end{对齐} \end{对齐}$$

我们证明\((**)\)（用“ \(\ge \) ”代替“>”）在较弱的条件下成立。假设，而不是\((*)\)，就足够了

$$\begin{aligned} \sum _{k=1}^N a_k \frac{\sin (kt)}{k}>0 \quad (N=1,\ldots ,n; \, 0<t< \pi ), \end{对齐}$$

此外，\((**)\)对更大的区域有效，即\(x_1,\ldots ,x_m\in (0,\pi )\)。